Money
The gear for quality shaving is more expensive than the regular stuff you can buy at the drug store. It is a good investment, however, not only because of its better quality, but because a) the shaving soap or cream will last you ages, b) the razor will last a lifetime, and c) blades for double-edged razors are much much cheaper than shaving cartridges for Mach 3 and the like.

Razors
Some people swear by double-edged, or safety, razors, others swear by the straight-edge or cut-throat razor. I’ve been shaving with a safety razor for over two years now, and couldn’t be happier. I’ve recently bought a cut-throat, which I’m trying to learn to use, just for the sport.

I can recommend the safety razor I use, which has the reputation of being the best for beginners:
The Merkur Heavy-Duty Classic. It is pleasingly simple and well crafted, made out of stainless steel. It’s heaver than, say, a Mach 3, and this helps with your shave. The Merkur is built to last a lifetime.

You can get it at Amazon, or the stores mentioned above.

Blades
There are three big brands here: Merkur, Derby and Feather.
The Merkurs are the dullest, but are not a bad choice for beginners.
Derbys are sharper than Merkurs, and very reliable. A favorite for many.
Feathers are the sharpest. Very sharp. Avoid them until you know what you’re doing. Once you do, Feather may become your favorite brand. Personally I prefer Derby.

Blades are hard to find, and the best choice is often to buy online in bulk. QED, linked in the section on stores, at the top of the page, is a good choice.

Cream or Soap
Real shaving cream or soap is much better than canned gel or foam. It requires a tiny bit of work on your part, but it’s enjoyable work. Either with your hands, or a brush, you need to mix the soap or cream with water, to form warm lather over your beard and mustache. Note that you don’t need a Santa Claus white beard of lather, like they show you in commercials for foam. The thing you’re trying to get is lubrication of the skin, plus soaking of the hairs to soften them. A thick foam doesn’t achieve this any better than a thin one.

As to brands, there are many, and you can find recommendations in Badger and Blade, but here are a few ideas.
Two cheap and good creams that are meant to be applied without a brush: Proraso and Musgo Real.
Great soap: Tabac.
The one American cream everyone loves (and the only American product on this page): Nancy Boy.
English brands that make good, expensive, creams and soaps: Trumper’s, Taylor’s, Truefit.

Brush, no brush
Some people prefer to apply the soap directly with their hands, but for most, using a brush gives better results and is more enjoyable.
There are two kinds of brush, depending on what hair they’re made of. Boar bristle brushes are cheap, and have a certain springiness. Badger hair brushes are expensive, but they retain water better than boar brushes. There are different degrees of badger hair, and they get more and more expensive with quality. Some people spend large quantities of money on badger hair brushes that have particularly good artistry. Say, upwards of $200.

Personally, I think boar bristle brushes can be great. I have one of them, plus a small badger hair brush that also does a great job, and is ideal for travel. It cost around $60.

A word on technique
If you’re coming from Mach 3 or equivalent modern razors, be careful not to apply too much pressure in your first few shaves.

You should not need to apply almost any pressure. The weight of the razor should do almost all the work for you. A good way of achieving this is holding the razor from its far end, with the ends of your fingers. Grabbing the handle tightly is almost surely going to result in a rough shave.

You get a first class shave by shaving in stages. You should re-lather between stages, and you should never shave un-lathered areas.
In the first stage, you shave in the direction of hair growth (with the grain). In the second stage you shave across the grain, and lastly, you shave against the grain.
The last stage, done properly, will give you what in Badger and Blade is referred to as BBS shave – baby butt smooth.